Some time ago I asked a question on consecutive numbers represented integrally by an integral positive binary quadratic form. It has occurred to me that, instead, the Green-Tao theorem may include a result on arithmetic progressions represented by a positive binary form. So my question is whether that is the case, do we already know that a positive binary form represents arbitrarily long arithmetic progressions? These would be primes in this general setting, thus quite different from consecutive integers of course.

My main reference is
David A. Cox,
Primes of the form $x^2 + n y^2.$ He defines the Dirichlet density on page 169. Then he states the
Chebotarev Density Theorem (8.17) on page 170. Finally he gives the Dirichlet density of primes represented by a positive binary form on page 188, Theorem 9.12. EDIT::: Not difficult to state: with discriminant $ \Delta < 0$ and class number $ h(\Delta),$ if the form is ambiguous (such as the principal form) the Dirichlet density of the set of primes it represents is $$ \frac{1}{2 h(\Delta)},$$
while if the form is not ambiguous the Dirichlet density is
$$ \frac{1}{h(\Delta)}.$$ On page 190 he does the example $ \Delta = -56.$ Here $x^2 + 14 y^2$ represents a set of primes with Dirichlet density $1/8,$ while $2 x^2 + 7 y^2$ also gets density $1/8,$ but in the other genus $ 3 x^2 + 2 x y + 5 y^2 $ and $ 3 x^2 - 2 x y + 5 y^2 $ each represent the same set of primes with density $1/4.$ Note on page 195 we have Exercise 9.17, that the sum of these densities for any discriminant must be $1/2.$ A little fiddling, not mentioned in the book, shows that each genus (of a fixed discriminant $\Delta$) represents the same total density, something we really want because of the relationship between genera and arithmetic progressions of primes.

same-day EDIT: I looked up Dirichlet density on wikipedia. As relates to the earlier David Hansen answer I liked, wikipedia stops short of saying that the Dirichlet density of primes in an arithmetic progression is identical to the relative density. Personally, I cannot see how the relative density could be anything else, but that is just my opinion. So I think I am also asking for references that prove the relative density is equal to the Dirichlet density in some naturally-occurring situations.

Later on the same day EDIT: There is stronger language in this next wikipedia page, so I think we can conclude that for primes in an arithmetic progression the Dirichlet density and the relative density are equal, but I would still like a more substantial reference. I'm the nervous type. I worry.

1 Answer
1

Edit: {The answer to your question, "...do we already know that a positive binary form represents arbitrarily long arithmetic progressions?" is yes. See the second paragraph below.}

If the relative density exists, so does the Dirichlet density and they are equal. The converse is not true in general. For primes in a given arithmetic progression, both densities exist. See Lang's Algebraic Number Theory, Ch. VIII.4 and XV. Given those facts, one approach to the problem would be trying to show that the relative density of the set of primes represented by a positive binary quadratic form actually exists (I have no idea how hard this might be).

On the other hand, if you only want to know about a.p.'s of primes represented by a positive quadratic form, a better approach might be answering the question, "Does Green-Tao still hold for sets of primes with positive Dirichlet density?" The answer is yes since G-T only requires that the limsup of the relative density be positive, and positive Dirichlet density implies positive limsup (if the limsup were 0, the lim would be 0).

Dear Kevin, Jurgen Neukirch "Class Field Theory" points to Serre "A Course in Arithmetic," and on page 76 he says that the set of primes $p$ such that a fixed polynomial has a root $ \pmod p $ has a natural density, and refers to K. Prachar "Primzahlverteilung" chapter 5 section 7. By results in the Cox book, this means that the principal form $x^2 + n y^2$ or $ x^2 + x y + k y^2$ does represent a set of primes with a natural density, therefore equalling the Dirichlet density. And by the result on arithmetic progressions, a full genus of forms has a natural density. Getting there.
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Will JagyJun 15 '10 at 16:33